Summaries
G
.
A
.
Leonov
,
V
.
A
.
Yakubovich
,
P
.
E
.
Tovstik
,
S
.
K
.
Matveev
Academician of the Russian Academy of Sciences Nikita Fedorovich Morozov
A.K. Belyaev, A.A. Muzaev, D.A. Indeitsev
. Derivation of the dynamic boundary value pro
b
lem
and state equ
a
tions fo
r polarized thermo

elastic materials
The dynamic boundary value problem and the state equations for simple polarized thermo

elastic m
a
terials are shown to be derivable from the first and second laws of thermodynamics.
Yu.
M. Dah
.
l
On
the
Kolosov
expressi
ons
in
the
plane
problem
of
elasticity
.
Three different Kolosov expressions for stresses which allow us to determine stresses in an
ela
s
tic body are found.
S.A. Zegzhda, G.A. Sinilshchikova.
The propagation of a crack in a thin beam under impulse
loadin
g.
The dynamics of propagation of a crack in a thin beam under the impact action of pressure on
the crack edges is studied. The crack is supposed to propagate in the symmetry plane of the
beam. The length of the crack, as well as the transversal force and
the bending moment at the
crack tip, are considered as generalized Lagrangian coord
i
nates in the zeroth approximation.
Successive approximations are constructed by introducing additional coord
i
nates that make it
possible to consider a few first modes of b
ending vibration of a free beam dynamically and all
the rest

quasistatically. The algorithm of construction of these approximations is offered. The
results of numerical calcul
a
tion are presented. The convergence analysis of the given method of
successive
approximations is done. The dependence of dynamics of the crack propagation on
the loading parameters is invest
i
gated.
V.A. Kuzkin, A.M. Krivtsov
.
The simple model for analytical derivation of the equation of state
for ideal crystals.
The simple model f
or elementary cell of 1D crystal is considered. The model is based on the
consideration of particle motion in the potential well. The microscopic definitions of such ma
c
roscopic parameters as pressure, thermal energy, and specific volume are given. The ana
lytical
equation of state in indirect form, which is valid for the wide range of the thermal energy vari
a
tions, is obtained. It is shown that usage of the Mie

Grűneisen equation of state can give co
n
siderable error in case of strong tensile deformations.
G.A. Leonov
. The criteria of cycles existence in two

dimensional quadratic systems
The existence of bifurcation of simultaneously arising cycles in the vicinity of two equ
i
librium
states of two

dimensional quadratic systems with one scalar parameter chang
ing is proved.
V
.
G
.
Osmolovskii
.
Exact solutions of the problem of phase transitions in a one

dimensional
model case.
The solutions of the problem of phase transitions in the explicit form are obtained in one

dimensional model case with both positive an
d zero coefficient of a surface

tension. The d
e
pendence of equilibrium states on the param
e
ters of the problem is investigated.
P.E.Tovstik
. On the asymptotic nature of the approximate models of beams, plates and shells
By using some test examples the re
sults obtained by the approximate models of beams, plates
and shells based on the kinematical hypotheses of Bernoulli

Kirchhoff

Love and T
i
moshe
n
ko

Reissner are compared with the a
s
ymptotic solutions of 3D equations of the theory of ela
s
ticity for narro
w areas. The statical problems and the pro
b
lems of free vibrations for bodies
made of the lin
e
arly elastic orthotropic material are studied. The main attention is paid to the
cases in which the m
a
terial stiffness in the tangential directions is much large
r than its stiffness
in the transversal dire
c
tion.
Mathematics
N.P. Alekseeva.
The combinatorial analysis of two basic forms of hidden periodicity in cat
e
gorial sequences
Two kinds of mixing periodical components according to the Spencer

Brown laws o
f form are
considered. If the identical fragments of a periodic component are kept then we have a pen
e
trant (call) form, if the identical fra
g
ments break up then we have a co

penetrant (crossing)
form. The symptom analysis of hidden periodicity is used for
the penetrant form, and the order
asymmetry method is used for the co

penetrant form. In the symptom analysis the main co
m
ponents method and SSA are modified for finite geometries. The order asymmetry method is
the cluster analysis where the metrics be
tween two gradations characterizes a deviation from
periodicity in a subs
e
quence over these gradations.
D.Y
u
. Bugaychenko.
Model checking MASL specification of distributed real

time systems.
We present model checking algorithms for MASL specification of
distributed real

time sy
s
tems. The proposed algorithms use symbolic model checking approach by analogy with model
checking algorithms for branching

time temporal logic CTL and alternating

time temporal lo
g
ic ATL. For the fixed environment case the algorit
hm is polynomial

time in the specific
a
tion
length and sizes of the set of system states and actions. For the dynamic env
i
ronment case the
algorithm is polynomial

time in the model size, but it is exponential

time in the structure of e
n
v
i
ronment specificati
on.
A.Kh. Gelig, V.A. Muranov
.
Stabilization of two classes of nonlinear pulse

modulated systems
with delay.
The Lyapunov function of the quadratic form is used for an analytical synthesis of a robust st
a
bili
z
ing control for two classes of nonlinear puls
e

modulated systems described by functional

differential equations with a lagging argument.
I.M. Davydova, E.Ya. Fedoseeva
. Generalization of the Chinese Remainder Theorem.
The derivation of the sufficient conditions for integer solvability of a system $
Ax=r$ in the
case of the basic m
a
trix $A$ in terms of the submatrices permanents of $A$ is presented. In the
Chinese Remainder Theorem the matrix $A$ is a particular case of the basic matrix. The der
i
vation can be e
x
tended for the case when a propositional
formula which describes the sign
scheme of $A$ is the minimal unsatisfiable CNF.
N.K. Kosovsky, T.M.
Kosovskaya
.
On the number of steps for constructing a Boolean solution
of polynomial comparisons and systems of them.
By means of fixing a parameter som
e NP

complete problems, connected with the solving of
co
m
parisons (and non

comparisons) of arithmetical terms by prime modulo, are decomposed as
a union of an infinite set of problems for which polynomial

time algorithms are constructed.
N.K. Krivulin
. Ev
aluation of the growth rate of the state vector in a generalized linear stocha
s
tic
sy
s
tem
A second

order generalized linear stochastic dynamical system is considered. The entries of the
sy
s
tem matrix are assumed to be independent and exponentially distrib
uted. In order to evaluate
the mean growth rate of the system state vector, a sequence of one

dimensional probability di
s
tributions is introduced. Derivation of the limiting density function is reduced to solving an a
l
gebraic linear system. The density is
used for evaluation of the mean growth rate for the system
under study.
V.V.
Nekrutkin
.
Martingale characterization of P
ò
lya processes and sequences.
It is proved that a mixed Poisson process
t
is a P
ò
lya process iff there exists a non

degenerate
linear
transformation
t
b
t
a
t
t
such that
t
is
the ma
r
tingale. The analogous result
is valid for P
ò
lya sequences.
T.M.
Tovstik
.
Calculation of the discrepancy for the finite number of points in the unit
n

dimensional cube.
The algorithm of the d
iscrepancy calculation for the finite number of points in the unit $n$

dimensional cube [0,1)
n
is given. This algorithm is simple for programming. For
4
2
n
the
time of the discrepancy calculation by this algorithm is esse
n
tially less tha
n for P.Bundschun’s
and Y.Zhu's algorithms. For greater $n$ the choice of the a
l
gorithm depends on the number of
checked points.
Mechanics
S.M. Bauer, A.N. Mironov
.
The contact between a spherical shell and an ela
s
tic ring.
The problem of the contact b
etween an elastic thin whole spherical shell and an elastic ring is
considered. The ring is considered as a spherical shell of a smaller radius with the edges sy
m
metric to the equator. The integral equation of contact is derived. The Green function is o
b
ta
ined. The integral equ
a
tion is reduced to the boundary problem. The distribution of contact
loading is found.
I.D. Volkov, M.A. Grekov
. Green's functions for bonded dissimilar materials with a slightly
curved interface.
The 2

D model of two

component ela
stic composite with a slightly curved interface is consi
d
ered. The conce
n
trated force and/or edge dislocation acts in one of the component. The solution
of both problems is represented in a form of series expansion of complex potentials in terms of
power o
f a small parameter on the base of the perturbation technique. The algorithm for deri
v
ing the co
m
plex potentials of any

order accurate solution in a closed form has been developed
for a wide range of local perturbed curves. Some basic formulas are given fo
r the first

order
solution when a local deviation of the interface from the straight one is described by power
functions. The stress distrib
u
tion at the curvilinear interface of a certain form is analyzed for
the case of concentrated force.
A.V. Mikheev
.
The influence of the shear parameter on the local stability of shallow orthotropic
shells on the ela
s
tic base
We consider the problem of buckling of a shallow orthotropic shell on the elastic base. The
equations of st
a
bility and the expressions of load pa
rameter are obtained. The critical loads for
the Kirchgoff

Love and Timoshenko models are compared. The results obtained are illustrated
by the case of a shell made of glass

fiber mat
e
rial.
V
.
S
.
Novoselov
.
The optimal two

impulse tangential passage with
the given relative velo
c
ity.
The variation method of the optimization of the coplanar two

impulse tangential passage with
the given relative velocity is proposed. The analytical construction of two successive approx
i
mations in the pro
b
lem of passage for b
oundary orbits with small eccentricities is given.
Astronomy
A.A. Bashakov, N.P. Pityev
.
Constructing self

consistent models of stellar systems by Schwar
z
schild's method.
The set of programs for constructing self

gravitating models of stellar systems in
a given distr
i
bution function of density using Schwarzschild's method is developed. As a trial gravitational
pote
n
tial the two

component model by Kutuzov and Ossipkov has been considered for which
the set of phase models and kinematic parameters have been
found.
K.V. Kholshevnikov
. Representation of gravity potential gradient of celestial bodies by means
of series in solid spherical harmonics.
It is well known that the derivative of a solid spherical harmonic of degree
n
is a solid spher
i
cal
harmonic of d
egree
n+1
. Using the algorithm by L.Cunningham
[1]
we succeed to expand the
derivative's coefficients as linear combinations of the original function's coeff
i
cients. It seems
to be the best expression for the gradient of the gravitational potential of any
celestial body re
p
resented by the Laplace series in spherical harmonics. Generalization to higher deriv
a
tives is
done. Similar procedure is performed with solid spherical harmonics, regular at the or
i
gin.
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